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## G = C2.(C4×Q16)  order 128 = 27

### 6th central stem extension by C2 of C4×Q16

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2.(C4×Q16)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4⋊C4 — C4×C4⋊C4 — C2.(C4×Q16)
 Lower central C1 — C2 — C2×C4 — C2.(C4×Q16)
 Upper central C1 — C23 — C2×C42 — C2.(C4×Q16)
 Jennings C1 — C2 — C2 — C22×C4 — C2.(C4×Q16)

Generators and relations for C2.(C4×Q16)
G = < a,b,c,d | a2=b4=c8=1, d2=c4, cbc-1=dbd-1=ab=ba, ac=ca, ad=da, dcd-1=ac-1 >

Subgroups: 244 in 127 conjugacy classes, 58 normal (44 characteristic)
C1, C2 [×7], C4 [×4], C4 [×10], C22 [×7], C8 [×3], C2×C4 [×6], C2×C4 [×2], C2×C4 [×20], Q8 [×6], C23, C42 [×4], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×5], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×2], C2×Q8 [×5], C2.C42 [×3], Q8⋊C4 [×4], Q8⋊C4 [×2], C2.D8 [×2], C2×C42, C2×C42, C2×C4⋊C4 [×3], C22×C8 [×2], C22×Q8, C22.7C42, C22.4Q16, C4×C4⋊C4, C23.67C23, C2×Q8⋊C4 [×2], C2×C2.D8, C2.(C4×Q16)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, Q16 [×2], C22×C4, C2×D4 [×2], C4○D4 [×4], C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, C2×Q16, C4○D8, C8⋊C22, C8.C22, C24.C22, C4×Q16, SD16⋊C4, C42Q16, D4.2D4, C23.48D4, C23.20D4, C2.(C4×Q16)

Smallest permutation representation of C2.(C4×Q16)
Regular action on 128 points
Generators in S128
(1 128)(2 121)(3 122)(4 123)(5 124)(6 125)(7 126)(8 127)(9 107)(10 108)(11 109)(12 110)(13 111)(14 112)(15 105)(16 106)(17 28)(18 29)(19 30)(20 31)(21 32)(22 25)(23 26)(24 27)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)(49 60)(50 61)(51 62)(52 63)(53 64)(54 57)(55 58)(56 59)(65 118)(66 119)(67 120)(68 113)(69 114)(70 115)(71 116)(72 117)(73 88)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)(89 100)(90 101)(91 102)(92 103)(93 104)(94 97)(95 98)(96 99)
(1 30 114 78)(2 20 115 86)(3 32 116 80)(4 22 117 88)(5 26 118 74)(6 24 119 82)(7 28 120 76)(8 18 113 84)(9 98 52 39)(10 96 53 46)(11 100 54 33)(12 90 55 48)(13 102 56 35)(14 92 49 42)(15 104 50 37)(16 94 51 44)(17 67 83 126)(19 69 85 128)(21 71 87 122)(23 65 81 124)(25 72 73 123)(27 66 75 125)(29 68 77 127)(31 70 79 121)(34 110 101 58)(36 112 103 60)(38 106 97 62)(40 108 99 64)(41 111 91 59)(43 105 93 61)(45 107 95 63)(47 109 89 57)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)
(1 62 5 58)(2 50 6 54)(3 60 7 64)(4 56 8 52)(9 117 13 113)(10 71 14 67)(11 115 15 119)(12 69 16 65)(17 40 21 36)(18 45 22 41)(19 38 23 34)(20 43 24 47)(25 35 29 39)(26 48 30 44)(27 33 31 37)(28 46 32 42)(49 126 53 122)(51 124 55 128)(57 121 61 125)(59 127 63 123)(66 109 70 105)(68 107 72 111)(73 102 77 98)(74 90 78 94)(75 100 79 104)(76 96 80 92)(81 101 85 97)(82 89 86 93)(83 99 87 103)(84 95 88 91)(106 118 110 114)(108 116 112 120)

G:=sub<Sym(128)| (1,128)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,105)(16,106)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59)(65,118)(66,119)(67,120)(68,113)(69,114)(70,115)(71,116)(72,117)(73,88)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)(89,100)(90,101)(91,102)(92,103)(93,104)(94,97)(95,98)(96,99), (1,30,114,78)(2,20,115,86)(3,32,116,80)(4,22,117,88)(5,26,118,74)(6,24,119,82)(7,28,120,76)(8,18,113,84)(9,98,52,39)(10,96,53,46)(11,100,54,33)(12,90,55,48)(13,102,56,35)(14,92,49,42)(15,104,50,37)(16,94,51,44)(17,67,83,126)(19,69,85,128)(21,71,87,122)(23,65,81,124)(25,72,73,123)(27,66,75,125)(29,68,77,127)(31,70,79,121)(34,110,101,58)(36,112,103,60)(38,106,97,62)(40,108,99,64)(41,111,91,59)(43,105,93,61)(45,107,95,63)(47,109,89,57), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,62,5,58)(2,50,6,54)(3,60,7,64)(4,56,8,52)(9,117,13,113)(10,71,14,67)(11,115,15,119)(12,69,16,65)(17,40,21,36)(18,45,22,41)(19,38,23,34)(20,43,24,47)(25,35,29,39)(26,48,30,44)(27,33,31,37)(28,46,32,42)(49,126,53,122)(51,124,55,128)(57,121,61,125)(59,127,63,123)(66,109,70,105)(68,107,72,111)(73,102,77,98)(74,90,78,94)(75,100,79,104)(76,96,80,92)(81,101,85,97)(82,89,86,93)(83,99,87,103)(84,95,88,91)(106,118,110,114)(108,116,112,120)>;

G:=Group( (1,128)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,107)(10,108)(11,109)(12,110)(13,111)(14,112)(15,105)(16,106)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27)(33,47)(34,48)(35,41)(36,42)(37,43)(38,44)(39,45)(40,46)(49,60)(50,61)(51,62)(52,63)(53,64)(54,57)(55,58)(56,59)(65,118)(66,119)(67,120)(68,113)(69,114)(70,115)(71,116)(72,117)(73,88)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87)(89,100)(90,101)(91,102)(92,103)(93,104)(94,97)(95,98)(96,99), (1,30,114,78)(2,20,115,86)(3,32,116,80)(4,22,117,88)(5,26,118,74)(6,24,119,82)(7,28,120,76)(8,18,113,84)(9,98,52,39)(10,96,53,46)(11,100,54,33)(12,90,55,48)(13,102,56,35)(14,92,49,42)(15,104,50,37)(16,94,51,44)(17,67,83,126)(19,69,85,128)(21,71,87,122)(23,65,81,124)(25,72,73,123)(27,66,75,125)(29,68,77,127)(31,70,79,121)(34,110,101,58)(36,112,103,60)(38,106,97,62)(40,108,99,64)(41,111,91,59)(43,105,93,61)(45,107,95,63)(47,109,89,57), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128), (1,62,5,58)(2,50,6,54)(3,60,7,64)(4,56,8,52)(9,117,13,113)(10,71,14,67)(11,115,15,119)(12,69,16,65)(17,40,21,36)(18,45,22,41)(19,38,23,34)(20,43,24,47)(25,35,29,39)(26,48,30,44)(27,33,31,37)(28,46,32,42)(49,126,53,122)(51,124,55,128)(57,121,61,125)(59,127,63,123)(66,109,70,105)(68,107,72,111)(73,102,77,98)(74,90,78,94)(75,100,79,104)(76,96,80,92)(81,101,85,97)(82,89,86,93)(83,99,87,103)(84,95,88,91)(106,118,110,114)(108,116,112,120) );

G=PermutationGroup([(1,128),(2,121),(3,122),(4,123),(5,124),(6,125),(7,126),(8,127),(9,107),(10,108),(11,109),(12,110),(13,111),(14,112),(15,105),(16,106),(17,28),(18,29),(19,30),(20,31),(21,32),(22,25),(23,26),(24,27),(33,47),(34,48),(35,41),(36,42),(37,43),(38,44),(39,45),(40,46),(49,60),(50,61),(51,62),(52,63),(53,64),(54,57),(55,58),(56,59),(65,118),(66,119),(67,120),(68,113),(69,114),(70,115),(71,116),(72,117),(73,88),(74,81),(75,82),(76,83),(77,84),(78,85),(79,86),(80,87),(89,100),(90,101),(91,102),(92,103),(93,104),(94,97),(95,98),(96,99)], [(1,30,114,78),(2,20,115,86),(3,32,116,80),(4,22,117,88),(5,26,118,74),(6,24,119,82),(7,28,120,76),(8,18,113,84),(9,98,52,39),(10,96,53,46),(11,100,54,33),(12,90,55,48),(13,102,56,35),(14,92,49,42),(15,104,50,37),(16,94,51,44),(17,67,83,126),(19,69,85,128),(21,71,87,122),(23,65,81,124),(25,72,73,123),(27,66,75,125),(29,68,77,127),(31,70,79,121),(34,110,101,58),(36,112,103,60),(38,106,97,62),(40,108,99,64),(41,111,91,59),(43,105,93,61),(45,107,95,63),(47,109,89,57)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128)], [(1,62,5,58),(2,50,6,54),(3,60,7,64),(4,56,8,52),(9,117,13,113),(10,71,14,67),(11,115,15,119),(12,69,16,65),(17,40,21,36),(18,45,22,41),(19,38,23,34),(20,43,24,47),(25,35,29,39),(26,48,30,44),(27,33,31,37),(28,46,32,42),(49,126,53,122),(51,124,55,128),(57,121,61,125),(59,127,63,123),(66,109,70,105),(68,107,72,111),(73,102,77,98),(74,90,78,94),(75,100,79,104),(76,96,80,92),(81,101,85,97),(82,89,86,93),(83,99,87,103),(84,95,88,91),(106,118,110,114),(108,116,112,120)])

38 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4R 4S 4T 4U 4V 8A ··· 8H order 1 2 ··· 2 4 ··· 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 2 ··· 2 4 ··· 4 8 8 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C4 D4 D4 Q16 C4○D4 C4○D8 C8⋊C22 C8.C22 kernel C2.(C4×Q16) C22.7C42 C22.4Q16 C4×C4⋊C4 C23.67C23 C2×Q8⋊C4 C2×C2.D8 Q8⋊C4 C4⋊C4 C22×C4 C2×C4 C2×C4 C22 C22 C22 # reps 1 1 1 1 1 2 1 8 2 2 4 8 4 1 1

Matrix representation of C2.(C4×Q16) in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 13 0 0 0 0 0 0 13 0 0 0 0 0 0 13 8 0 0 0 0 13 4 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 0 11 0 0 0 0 3 0 0 0 0 0 0 0 1 15 0 0 0 0 0 16 0 0 0 0 0 0 6 11 0 0 0 0 3 0
,
 11 7 0 0 0 0 12 6 0 0 0 0 0 0 16 0 0 0 0 0 16 1 0 0 0 0 0 0 4 0 0 0 0 0 4 13

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,13,0,0,0,0,8,4,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,3,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,15,16,0,0,0,0,0,0,6,3,0,0,0,0,11,0],[11,12,0,0,0,0,7,6,0,0,0,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,4,4,0,0,0,0,0,13] >;

C2.(C4×Q16) in GAP, Magma, Sage, TeX

C_2.(C_4\times Q_{16})
% in TeX

G:=Group("C2.(C4xQ16)");
// GroupNames label

G:=SmallGroup(128,660);
// by ID

G=gap.SmallGroup(128,660);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,736,422,394,2804,718,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=1,d^2=c^4,c*b*c^-1=d*b*d^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=a*c^-1>;
// generators/relations

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